Applications of HCF and LCM

The same machinery we just built , HCF and LCM via prime factorisation , solves dozens of practical questions. Board examiners love these because a clean translation from English to mathematics is half the battle.

Two big questions

Most word problems in this chapter ask one of two questions:

"What is the largest size / measure / interval that divides all the given quantities exactly?" This is an HCF question.
"After how long / how many units do all the given cycles synchronise again?" This is an LCM question.

A quick test: if the answer must divide the given numbers, you want HCF. If the answer must be a multiple of the given numbers, you want LCM.

Theorem in disguise

The product rule $\operatorname{HCF}(a,b) \cdot \operatorname{LCM}(a,b) = a \cdot b$ shows up constantly. Whenever a problem gives you the HCF, the LCM, and one of the numbers, you can immediately find the other: $b = \frac{a \cdot \operatorname{LCM}(a,b)}{\operatorname{HCF}(a,b)} \div \frac{a}{a} = \frac{\operatorname{HCF}(a,b) \cdot \operatorname{LCM}(a,b)}{a}.$ Equivalent and simpler: $a \cdot b = \operatorname{HCF} \cdot \operatorname{LCM}$ , so $b = \dfrac{\operatorname{HCF} \cdot \operatorname{LCM}}{a}$ .

A second pattern: if the HCF is $h$ and we are told two numbers are $ah$ and $bh$ with $\gcd(a, b) = 1$ , then the LCM is $abh$ . So if a question states "two numbers in ratio $p : q$ (in lowest terms) with HCF $h$ ", the numbers are $ph$ and $qh$ , and their LCM is $pqh$ .

A third: if we are looking for the largest piece (HCF) but the problem gives leftover remainders ( $r_1$ , $r_2$ , $\ldots$ ), subtract the remainders first. The greatest number that divides $a$ leaving remainder $r$ is the same as the greatest number that divides $a - r$ .

The fourth pattern is "smallest multiple plus shift". If we want the smallest number that leaves remainder $r$ when divided by each of $a, b, c$ , that smallest number is $\operatorname{LCM}(a,b,c) + r$ .

Worked examples

Example 1. Find the largest number that divides $245$ and $1029$ , leaving remainders $5$ and $5$ respectively.

The "largest number" wants HCF. Subtract the remainders: we need HCF of $245 - 5 = 240$ and $1029 - 5 = 1024$ .

$240 = 2^4 \times 3 \times 5$ , $1024 = 2^{10}$ . Common primes: $2$ with min exponent $4$ . So HCF $= 2^4 = 16$ .

The required number is $\boxed{16}$ .

Example 2. Find the smallest number which when divided by $35$ , $56$ , and $91$ leaves a remainder of $7$ in each case.

LCM-with-shift pattern. First find LCM $(35, 56, 91)$ .

$35 = 5 \times 7$ , $56 = 2^3 \times 7$ , $91 = 7 \times 13$ . Max exponents: $2^3, 5, 7, 13$ . LCM $= 8 \times 5 \times 7 \times 13 = 3640$ .

Required number $= 3640 + 7 = \boxed{3647}$ .

Example 3. Two tankers contain $850$ litres and $680$ litres of petrol. Find the maximum capacity of a container that can measure the petrol of either tanker exactly.

"Maximum capacity dividing both" = HCF.

$850 = 2 \times 5^2 \times 17$ , $680 = 2^3 \times 5 \times 17$ . Common primes: $2$ (min $1$ ), $5$ (min $1$ ), $17$ (min $1$ ). HCF $= 2 \times 5 \times 17 = 170$ litres.

Example 4. Three traffic signals at a junction change every $48$ s, $72$ s, and $108$ s. They start changing simultaneously at $7:00$ a.m. When will they next change simultaneously?

LCM $(48, 72, 108)$ .

$48 = 2^4 \times 3$ , $72 = 2^3 \times 3^2$ , $108 = 2^2 \times 3^3$ . Max exponents: $2^4, 3^3$ . LCM $= 16 \times 27 = 432$ s $= 7$ min $12$ s.

Next simultaneous change at $7$ h $7$ min $12$ s a.m.

Example 5. The HCF and LCM of two numbers are $9$ and $360$ . If one number is $45$ , find the other.

By the product rule, $a \cdot b = \operatorname{HCF} \cdot \operatorname{LCM} = 9 \times 360 = 3240$ . So $b = 3240 / 45 = 72$ .

Try it yourself

Find the largest number that divides $1251$ and $9377$ leaving remainders $1$ and $2$ respectively.
Find the smallest number which when divided by $24$ , $36$ , and $54$ leaves a remainder of $5$ each time.
Two rods are $1.20$ m and $1.80$ m long. They are to be cut into equal pieces of the greatest possible length. Find that length and the total number of pieces.
The HCF and LCM of two numbers are $4$ and $9696$ . If one number is $96$ , find the other.
A merchant has $120$ litres of one oil and $180$ litres of another. He wants to sell them in tins of equal capacity. What is the greatest capacity of such a tin?
Three bells chime every $9$ , $12$ , and $15$ minutes. If they all chime together at $7:00$ a.m., when will they next chime together?
Two numbers are in ratio $3:4$ and their LCM is $180$ . Find the numbers.
Find the smallest $5$ -digit number divisible by $12$ , $15$ , and $18$ .
Two friends start jogging in opposite directions around a circular track of $360$ m, with speeds giving lap times of $60$ s and $90$ s. When do they next meet at the start?
Show that the HCF of any two consecutive natural numbers is $1$ .

Pitfalls / Insight

"Largest that divides X, Y" → HCF; "smallest divisible by X, Y" → LCM. Wrong choice is the most common slip.
Remember to subtract remainders for HCF questions and add remainders for LCM questions.
Always factorise once and reuse. Don't recompute factorisations across sub-questions.

Insight. Word problems often hide their structure under one or two extra English sentences. Find the quantity that the answer must divide (HCF) or the quantity that the answer must be a multiple of (LCM), and the rest is arithmetic.